Galois group of \(X^ p+ aX+a\) (Q1911797)

In this paper the Galois group \(G\) over \(\mathbb{Q}\) of trinomials of the form \(f(X)= X^p+ aX+ a\) is studied, where \(p\) is a prime number and \(a\) is an integer such that \(p\) divides \(a\) exactly once (Eisenstein type trinomial with respect to \(p)\). The author determines the inertia group of \(p\) in \(N/ \mathbb{Q}\) and the splitting field of \(f(X)\), and he proves that the Galois group \(G\) is the symmetric group \(S_p\) or the group \(\text{Aff} (\mathbb{F}_p)\) of all affine transformations over \(\mathbb{F}_p\). Furthermore, he shows that \(G \simeq S_p\) in each of the two following cases: if \(a<0\) or if \(a/p \not\equiv 1 \pmod p\).